Deeply quasi-bound state in single- and double-$\bar{K}$ nuclear clusters

New calculations of the quasi-bound state positions in $K^{-}{K}^{-}pp$ kaonic nuclear cluster are performed using non-relativistic four-body Faddeev-type equations in AGS form. The corresponding separable approximation for the integral kernels in the three- and four-body kaonic clusters is obtained by using the Hilbert-Schmidt expansion procedure. Different phenomenological models of $\bar{K}N-\pi \mathrm{\Sigma }$ potentials with one- and two-pole structure of $\mathrm{\Lambda }$(1405) resonance and separable potential models for $\bar{K}$-$\bar{K}$ and nucleon-nucleon interactions, are used. The dependence of the resulting four-body binding energy on models of $\bar{K}N-\pi \mathrm{\Sigma }$ interaction is investigated. We obtained the binding energy of the $K^{-}{K}^{-}pp$ quasi-bound state $\sim$ 80-94 MeV with the phenomenological $\bar{K}N$ potentials. The width is about $\sim$ 5-8 MeV for the two-pole models of the interaction, while the one-pole potentials give $\sim$ 24-31 MeV width.